↑

Properties

Label 2-3525-705.563-c0-0-27
Degree $2$
Conductor $3525$
Sign $0.0787 + 0.996i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.147 − 0.147i)2-s + (0.998 + 0.0523i)3-s − 0.956i·4-s + (−0.139 − 0.155i)6-s + (0.575 − 0.575i)7-s + (−0.289 + 0.289i)8-s + (0.994 + 0.104i)9-s + (0.0500 − 0.954i)12-s − 0.170·14-s − 0.870·16-s + (−1.38 − 1.38i)17-s + (−0.131 − 0.162i)18-s + (0.604 − 0.544i)21-s + (−0.303 + 0.273i)24-s + (0.987 + 0.156i)27-s + (−0.550 − 0.550i)28-s + ⋯
L(s)  = 1  + (−0.147 − 0.147i)2-s + (0.998 + 0.0523i)3-s − 0.956i·4-s + (−0.139 − 0.155i)6-s + (0.575 − 0.575i)7-s + (−0.289 + 0.289i)8-s + (0.994 + 0.104i)9-s + (0.0500 − 0.954i)12-s − 0.170·14-s − 0.870·16-s + (−1.38 − 1.38i)17-s + (−0.131 − 0.162i)18-s + (0.604 − 0.544i)21-s + (−0.303 + 0.273i)24-s + (0.987 + 0.156i)27-s + (−0.550 − 0.550i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0787 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.0787 + 0.996i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.0787 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703579818\)
\(L(\frac12)\) \(\approx\) \(1.703579818\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.998 - 0.0523i)T \)
5 \( 1 \)
47 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.147 + 0.147i)T + iT^{2} \)
7 \( 1 + (-0.575 + 0.575i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (1.14 - 1.14i)T - iT^{2} \)
59 \( 1 - 0.813T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.48iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 0.618iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 - 1.17T + T^{2} \)
97 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.724966333766144999885422500804, −7.897276531578032893085320492529, −7.13685634885213740267842925418, −6.55627232129858799728251156208, −5.41324933515654562014546281674, −4.62523265036189399727560600334, −4.05144789652261300149149295743, −2.72571398094428916111818778414, −2.08151474883516564647455733143, −0.950402859859755222489448082116, 1.75899956699524387299759454868, 2.49712375382596215208945749905, 3.40316365284429834197600080353, 4.20475119629012802115176003473, 4.86424503234186242075491087943, 6.25137682680989031032416463665, 6.79032523945615737656590042491, 7.75009876397396978189170805805, 8.318676241628548876824213823698, 8.605112523325072846975940675619

Graph of the $Z$-function along the critical line